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The Finite Element Method:A Four-Article Series

FINITE ELEMENT ANALYSIS: Introductionby Steve Roensch, President, Roensch & Associates

...First in a four-part series

Finite element analysis (FEA) is a fairly recent discipline crossing the boundaries ofmathematics, physics, engineering and computer science. The method has wide applicationand enjoys extensive utilization in the structural, thermal and fluid analysis areas. The finiteelement method is comprised of three major phases: (1) pre-processing, in which theanalyst develops a finite element mesh to divide the subject geometry into subdomains formathematical analysis, and applies material properties and boundary conditions, (2)solution, during which the program derives the governing matrix equations from the modeland solves for the primary quantities, and (3) post-processing, in which the analyst checksthe validity of the solution, examines the values of primary quantities (such as displacementsand stresses), and derives and examines additional quantities (such as specialized stressesand error indicators).

The advantages of FEA are numerous and important. A new design concept may be modeledto determine its real world behavior under various load environments, and may therefore berefined prior to the creation of drawings, when few dollars have been committed and changesare inexpensive. Once a detailed CAD model has been developed, FEA can analyze thedesign in detail, saving time and money by reducing the number of prototypes required. Anexisting product which is experiencing a field problem, or is simply being improved, can beanalyzed to speed an engineering change and reduce its cost. In addition, FEA can beperformed on increasingly affordable computer workstations and personal computers, andprofessional assistance is available.

It is also important to recognize the limitations of FEA. Commercial software packages andthe required hardware, which have seen substantial price reductions, still require a significantinvestment. The method can reduce product testing, but cannot totally replace it. Probably

most important, an inexperienced user can deliver incorrect answers, upon which expensivedecisions will be based. FEA is a demanding tool, in that the analyst must be proficient notonly in elasticity or fluids, but also in mathematics, computer science, and especially thefinite element method itself.

Which FEA package to use is a subject that cannot possibly be covered in this shortdiscussion, and the choice involves personal preferences as well as package functionality.Where to run the package depends on the type of analyses being performed. A typical finiteelement solution requires a fast, modern disk subsystem for acceptable performance.Memory requirements are of course dependent on the code, but in the interest ofperformance, the more the better, with a representative range measured in gigabytes per

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performance, the more the better, with a representative range measured in gigabytes peruser. Processing power is the final link in the performance chain, with clock speed, cache,pipelining and multi-processing all contributing to the bottom line. These analyses can run forhours on the fastest systems, so computing power is of the essence.

One aspect often overlooked when entering the finite element area is education. Withoutadequate training on the finite element method and the specific FEA package, a new user willnot be productive in a reasonable amount of time, and may in fact fail miserably. Expect todedicate one to two weeks up front, and another one to two weeks over the first year, to eitherclassroom or self-help education. It is also important that the user have a basic understandingof the computer's operating system.

Next month's article will go into detail on the pre-processing phase of the finite elementmethod.

© 2008-2013 Roensch & Associates. All rights reserved.

1. Introduction

2. Pre-processing

3. Solution

4. Post-processing

This four-article series was published in a newsletter of the American Society ofMechanical Engineers (ASME). It serves as an introduction to the recent analysisdiscipline known as the finite element method. The author is an engineeringconsultant and expert witness specializing in finite element analysis.

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FINITE ELEMENT ANALYSIS: Pre-processingby Steve Roensch, President, Roensch & Associates

...Second in a four-part series

As discussed last month, finite element analysis is comprised of pre-processing, solution andpost-processing phases. The goals of pre-processing are to develop an appropriate finiteelement mesh, assign suitable material properties, and apply boundary conditions in the formof restraints and loads.

The finite element mesh subdivides the geometry into elements, upon which are foundnodes. The nodes, which are really just point locations in space, are generally located at theelement corners and perhaps near each midside. For a two-dimensional (2D) analysis, or athree-dimensional (3D) thin shell analysis, the elements are essentially 2D, but may be"warped" slightly to conform to a 3D surface. An example is the thin shell linear quadrilateral;thin shell implies essentially classical shell theory, linear defines the interpolation ofmathematical quantities across the element, and quadrilateral describes the geometry. For a3D solid analysis, the elements have physical thickness in all three dimensions. Commonexamples include solid linear brick and solid parabolic tetrahedral elements. In addition, thereare many special elements, such as axisymmetric elements for situations in which thegeometry, material and boundary conditions are all symmetric about an axis.

The model's degrees of freedom (dof) are assigned at the nodes. Solid elements generallyhave three translational dof per node. Rotations are accomplished through translations ofgroups of nodes relative to other nodes. Thin shell elements, on the other hand, have six dofper node: three translations and three rotations. The addition of rotational dof allows forevaluation of quantities through the shell, such as bending stresses due to rotation of onenode relative to another. Thus, for structures in which classical thin shell theory is a validapproximation, carrying extra dof at each node bypasses the necessity of modeling thephysical thickness. The assignment of nodal dof also depends on the class of analysis. For a

thermal analysis, for example, only one temperature dof exists at each node.

Developing the mesh is usually the most time-consuming task in FEA. In the past, nodelocations were keyed in manually to approximate the geometry. The more modern approachis to develop the mesh directly on the CAD geometry, which will be (1) wireframe, with pointsand curves representing edges, (2) surfaced, with surfaces defining boundaries, or (3) solid,defining where the material is. Solid geometry is preferred, but often a surfacing package cancreate a complex blend that a solids package will not handle. As far as geometric detail, anunderlying rule of FEA is to "model what is there", and yet simplifying assumptions simplymust be applied to avoid huge models. Analyst experience is of the essence.

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The geometry is meshed with a mapping algorithm or an automatic free-meshing algorithm.The first maps a rectangular grid onto a geometric region, which must therefore have thecorrect number of sides. Mapped meshes can use the accurate and cheap solid linear brick3D element, but can be very time-consuming, if not impossible, to apply to complexgeometries. Free-meshing automatically subdivides meshing regions into elements, with theadvantages of fast meshing, easy mesh-size transitioning (for a denser mesh in regions oflarge gradient), and adaptive capabilities. Disadvantages include generation of huge models,generation of distorted elements, and, in 3D, the use of the rather expensive solid parabolictetrahedral element. It is always important to check elemental distortion prior to solution. Abadly distorted element will cause a matrix singularity, killing the solution. A less distortedelement may solve, but can deliver very poor answers. Acceptable levels of distortion aredependent upon the solver being used.

Material properties required vary with the type of solution. A linear statics analysis, forexample, will require an elastic modulus, Poisson's ratio and perhaps a density for eachmaterial. Thermal properties are required for a thermal analysis. Examples of restraints aredeclaring a nodal translation or temperature. Loads include forces, pressures and heat flux. Itis preferable to apply boundary conditions to the CAD geometry, with the FEA packagetransferring them to the underlying model, to allow for simpler application of adaptive andoptimization algorithms. It is worth noting that the largest error in the entire process is often inthe boundary conditions. Running multiple cases as a sensitivity analysis may be required.

Next month's article will discuss the solution phase of the finite element method.


1. Introduction

2. Pre-processing

3. Solution

4. Post-processing

This four-article series was published in a newsletter of the American Society ofMechanical Engineers (ASME). It serves as an introduction to the recent analysis

discipline known as the finite element method. The author is an engineeringconsultant and expert witness specializing in finite element analysis.

262-375-2228 Courtroom FEA and the static and animated meshed Rare trademarks of Roensch & Associates.© 2013 Roensch & Associates. All rights reserved.














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FINITE ELEMENT ANALYSIS: Solutionby Steve Roensch, President, Roensch & Associates

...Third in a four-part series

While the pre-processing and post-processing phases of the finite element method areinteractive and time-consuming for the analyst, the solution is often a batch process, and isdemanding of computer resource. The governing equations are assembled into matrix formand are solved numerically. The assembly process depends not only on the type of analysis(e.g. static or dynamic), but also on the model's element types and properties, materialproperties and boundary conditions.

In the case of a linear static structural analysis, the assembled equation is of the form Kd =r, where K is the system stiffness matrix, d is the nodal degree of freedom (dof) displacementvector, and r is the applied nodal load vector. To appreciate this equation, one must beginwith the underlying elasticity theory. The strain-displacement relation may be introduced intothe stress-strain relation to express stress in terms of displacement. Under the assumption ofcompatibility, the differential equations of equilibrium in concert with the boundary conditionsthen determine a unique displacement field solution, which in turn determines the strain andstress fields. The chances of directly solving these equations are slim to none for anythingbut the most trivial geometries, hence the need for approximate numerical techniquespresents itself.

A finite element mesh is actually a displacement-nodal displacement relation, which, throughthe element interpolation scheme, determines the displacement anywhere in an elementgiven the values of its nodal dof. Introducing this relation into the strain-displacement relation,we may express strain in terms of the nodal displacement, element interpolation scheme anddifferential operator matrix. Recalling that the expression for the potential energy of an elasticbody includes an integral for strain energy stored (dependent upon the strain field) andintegrals for work done by external forces (dependent upon the displacement field), we can

therefore express system potential energy in terms of nodal displacement.

Applying the principle of minimum potential energy, we may set the partial derivative ofpotential energy with respect to the nodal dof vector to zero, resulting in: a summation ofelement stiffness integrals, multiplied by the nodal displacement vector, equals a summationof load integrals. Each stiffness integral results in an element stiffness matrix, which sum toproduce the system stiffness matrix, and the summation of load integrals yields the appliedload vector, resulting in Kd = r. In practice, integration rules are applied to elements, loadsappear in the r vector, and nodal dof boundary conditions may appear in the d vector or maybe partitioned out of the equation.

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Solution methods for finite element matrix equations are plentiful. In the case of the linearstatic Kd = r, inverting K is computationally expensive and numerically unstable. A bettertechnique is Cholesky factorization, a form of Gauss elimination, and a minor variation on the"LDU" factorization theme. The K matrix may be efficiently factored into LDU, where L islower triangular, D is diagonal, and U is upper triangular, resulting in LDUd = r. Since L and Dare easily inverted, and U is upper triangular, d may be determined by back-substitution.Another popular approach is the wavefront method, which assembles and reduces theequations at the same time. Some of the best modern solution methods employ sparsematrix techniques. Because node-to-node stiffnesses are non-zero only for nearby node pairs,the stiffness matrix has a large number of zero entries. This can be exploited to reducesolution time and storage by a factor of 10 or more. Improved solution methods arecontinually being developed. The key point is that the analyst must understand the solutiontechnique being applied.

Dynamic analysis for too many analysts means normal modes. Knowledge of the naturalfrequencies and mode shapes of a design may be enough in the case of a single-frequencyvibration of an existing product or prototype, with FEA being used to investigate the effects ofmass, stiffness and damping modifications. When investigating a future product, or anexisting design with multiple modes excited, forced response modeling should be used toapply the expected transient or frequency environment to estimate the displacement and evendynamic stress at each time step.

This discussion has assumed h-code elements, for which the order of the interpolationpolynomials is fixed. Another technique, p-code, increases the order iteratively untilconvergence, with error estimates available after one analysis. Finally, the boundary elementmethod places elements only along the geometrical boundary. These techniques havelimitations, but expect to see more of them in the near future.

Next month's article will discuss the post-processing phase of the finite element method.


1. Introduction

2. Pre-processing

3. Solution

4. Post-processing
















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FINITE ELEMENT ANALYSIS: Post-processingby Steve Roensch, President, Roensch & Associates

...Last in a four-part series

After a finite element model has been prepared and checked, boundary conditions have beenapplied, and the model has been solved, it is time to investigate the results of the analysis.This activity is known as the post-processing phase of the finite element method.

Post-processing begins with a thorough check for problems that may have occurred duringsolution. Most solvers provide a log file, which should be searched for warnings or errors, andwhich will also provide a quantitative measure of how well-behaved the numerical procedureswere during solution. Next, reaction loads at restrained nodes should be summed andexamined as a "sanity check". Reaction loads that do not closely balance the applied loadresultant for a linear static analysis should cast doubt on the validity of other results. Errornorms such as strain energy density and stress deviation among adjacent elements might belooked at next, but for h-code analyses these quantities are best used to target subsequentadaptive remeshing.

Once the solution is verified to be free of numerical problems, the quantities of interest maybe examined. Many display options are available, the choice of which depends on themathematical form of the quantity as well as its physical meaning. For example, thedisplacement of a solid linear brick element's node is a 3-component spatial vector, and themodel's overall displacement is often displayed by superposing the deformed shape over theundeformed shape. Dynamic viewing and animation capabilities aid greatly in obtaining anunderstanding of the deformation pattern. Stresses, being tensor quantities, currently lack agood single visualization technique, and thus derived stress quantities are extracted anddisplayed. Principal stress vectors may be displayed as color-coded arrows, indicating bothdirection and magnitude. The magnitude of principal stresses or of a scalar failure stress suchas the Von Mises stress may be displayed on the model as colored bands. When this type of

display is treated as a 3D object subjected to light sources, the resulting image is known asa shaded image stress plot. Displacement magnitude may also be displayed by coloredbands, but this can lead to misinterpretation as a stress plot.

An area of post-processing that is rapidly gaining popularity is that of adaptive remeshing.Error norms such as strain energy density are used to remesh the model, placing a densermesh in regions needing improvement and a coarser mesh in areas of overkill. Adaptivityrequires an associative link between the model and the underlying CAD geometry, and worksbest if boundary conditions may be applied directly to the geometry, as well. Adaptiveremeshing is a recent demonstration of the iterative nature of h-code analysis.

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Optimization is another area enjoying recent advancement. Based on the values of variousresults, the model is modified automatically in an attempt to satisfy certain performancecriteria and is solved again. The process iterates until some convergence criterion is met. Inits scalar form, optimization modifies beam cross-sectional properties, thin shell thicknessesand/or material properties in an attempt to meet maximum stress constraints, maximumdeflection constraints, and/or vibrational frequency constraints. Shape optimization is morecomplex, with the actual 3D model boundaries being modified. This is best accomplished byusing the driving dimensions as optimization parameters, but mesh quality at each iterationcan be a concern.

Another direction clearly visible in the finite element field is the integration of FEA packageswith so-called "mechanism" packages, which analyze motion and forces of large-displacement multi-body systems. A long-term goal would be real-time computation anddisplay of displacements and stresses in a multi-body system undergoing large displacementmotion, with frictional effects and fluid flow taken into account when necessary. It is difficult toestimate the increase in computing power necessary to accomplish this feat, but 2 or 3orders of magnitude is probably close. Algorithms to integrate these fields of analysis may beexpected to follow the computing power increases.

In summary, the finite element method is a relatively recent discipline that has quicklybecome a mature method, especially for structural and thermal analysis. The costs ofapplying this technology to everyday design tasks have been dropping, while the capabilitiesdelivered by the method expand constantly. With education in the technique and in thecommercial software packages becoming more and more available, the question has movedfrom "Why apply FEA?" to "Why not?". The method is fully capable of delivering higherquality products in a shorter design cycle with a reduced chance of field failure, provided it isapplied by a capable analyst. It is also a valid indication of thorough design practices, shouldan unexpected litigation crop up. The time is now for industry to make greater use of this andother analysis techniques.


1. Introduction

2. Pre-processing

3. Solution

4. Post-processing















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